Problem 45 - A Graphical Approach to Precalculus with Limits

Question

Graph \(y=\log x^{3}\) and \(y=3 \log x\) on separate sets of axes. It would seem, at first glance, that by applying the power rule for logarithms, these graphs should be the same. Are they? If not, why not? (Hint: Consider the domain in each case.)

Step-by-Step Solution

Answer

The graphs are the same for \(x > 0\), justified by domain matching through power rules and behavioral analysis.

Step 1: Analyze the Expression \(y = \log x^3\)

The expression \(y = \log x^3\) can be rewritten using the logarithm power rule as \(y = 3\log x\). This suggests that its behavior should match \(y = 3\log x\). However, rewriting it in this way assumes that \(x^3 > 0\), meaning \(x eq 0\). Furthermore, for a real log function to be defined, \(x > 0\). Therefore, the domain of \(y = \log x^3\) is \(x > 0\).

Step 2: Analyze the Expression \(y = 3 \log x\)

The expression \(y = 3\log x\) directly reveals that the log function is defined when \(x > 0\). Thus, the domain of \(y = 3 \log x\) is also \(x > 0\).

Step 3: Compare the Graphs and Domains

Given both functions appear to suggest the same relationship, \(3\log x\), they indeed seem similar graphically, lying on the same curve where \(x > 0\). Both graphs coincide for positive \(x\). However, if the calculation of \(x^3\) extends to considering when it is negative (i.e., negative bases raised to an odd power, conceptual in some contexts), the domain for such consideration would only exist theoretically for real logs; by typical real-world standards, complex numbers utilize rules not accounted for here.

Step 4: Conclusion

Despite similar algebraic manipulations and results within their shared domain \(x > 0\), the context of the power law strictly limits \(\log x^3\)'s two sides to matching the real domain of the simpler \(3 \log x\). Thus, while in real analytical terms they mirror each other over explicitly positive values, they are considered equivalent in existing practical applications.

Key Concepts

Graphing Logarithms

When graphing logarithmic functions, it's essential to understand their curves and how they behave. For instance, if we take the function \(y = \log x^3\) and rewrite it using the power rule, it transforms into \(y = 3\log x\).

To graph these functions accurately, we follow these steps:

• Identify the domain: Only positive values of \(x\) are considered for real logarithms.
• Determine key points: For example, at \(x = 1\), \(\log 1 = 0\), hence both graphs will pass through \(y = 0\).
• Analyze the behavior: As \(x\) increases, the value of \(\log x\) increases, thus scaling these values by three (in \(3\log x\)) gives a steeper graph.

The tricky part is understanding that while these transformed functions seem similar, small nuances in understanding domains can alter their true representation.

Power Rule for Logarithms

The power rule for logarithms is a handy mathematical property that simplifies expressions. It states that the logarithm of a power can be expressed as the exponent multiplied by the logarithm of the base. Formally, \(\log x^n = n \log x\).

Using this rule allows us to simplify and manipulate logarithmic expressions.

• Simplification: \( \log x^3 = 3\log x \).
• Computation ease: Reduces complex expressions, making calculations manageable.
• Clarification: Clearly denotes how repeated multiplication in exponents impacts the log scale.

However, it's crucial to remember that this rule applies within the domain of definable logarithms. Both expressions \(y = \log x^3\) and \(y = 3 \log x\) rely on \(x > 0\) to remain valid. Real solutions adhere strictly to the domain restrictions.

Domain of a Function

Understanding the domain is fundamental when dealing with functions, especially with logarithms. The domain of a function defines all the possible input values (\(x\) values) for which the function is defined and yields real numbers. For logarithmic functions, this generally means the argument of the logarithm must be greater than zero.

In our equations:

• For \(y = \log x^3\), we need \(x^3 > 0\). Thus, \(x > 0\).
• For \(y = 3\log x\), the condition is simply \(x > 0\) since \(\log x\) needs \(x\) to be positive.

These imply that both functions share the same domain, \(x > 0\), which must be respected to ensure valid and real results. In practice, understanding the domain helps in avoiding scenarios where the function becomes undefined, especially when graphing or computing solutions.